KS3 Maths · Year 7 · Chapter 1: Working with numbers · Lesson 3

Multiples and lowest common multiple

Year 7About 60 minutesNo calculator49 marks

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Quick questions to start.

A1
Needed today
Write down the first five multiples of $7$ .
[1 mark]
Answer
A2
Needed today
Write $\frac{3}{5}$ as a fraction with denominator $20$ .
[1 mark]
Answer
A3
Last lesson
Find the HCF of $24$ and $40$ .
[1 mark]
Answer
A4
From chapter 1
Work out $- 9 \times - 6$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find the lowest common multiple (LCM) of $12$ and $18$ .

Hint List the multiples of each number until the same number appears in both lists.

B1
Find the lowest common multiple (LCM) of $14$ and $35$ .
[2 marks]
Answer

Example 2

Find the LCM of $4$ , $6$ and $10$ .

Hint List the multiples of the largest number,

10

. Stop at the first one that

4

and

6

both divide.

B2
Find the LCM of $6$ , $9$ and $15$ .
[2 marks]
Answer

Example 3

Work out $\frac{5}{6} + \frac{4}{9}$ . Give your answer as a mixed number.

Hint Use the LCM of

6

and

8

as the common denominator.

B3
Work out
(a) $\frac{7}{12} + \frac{2}{15}$
(b) $\frac{5}{8} - \frac{1}{6}$
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

Two lighthouses flash their lights. One flashes every $20$ seconds and the other every $45$ seconds. They flash at the same time at 9:00 pm. At what time do they next flash at the same time?

Hint They flash together again after a number of seconds that is a multiple of

20

and of

45

B4
From a bus station, a number 7 bus leaves every $16$ minutes and a number 9 bus leaves every $24$ minutes. Both buses leave at 8:10 am. At what time do they next leave together?
[3 marks]
Answer

Example 5

Two numbers have an HCF of $6$ and an LCM of $36$ . Neither number is $6$ or $36$ . Find the two numbers.

Hint Both numbers are multiples of

6

and factors of

36

B5
Two numbers have an HCF of $4$ and an LCM of $60$ . Neither number is $4$ or $60$ . Find the two numbers.
[2 marks]
Answer

Practice

Year 7

C1
The pupils in Year 7 can be split into equal groups of $8$ , of $10$ or of $12$ , with nobody left over. Work out the smallest number of pupils there could be in Year 7.
[2 marks]
Answer
C2
Find the HCF and the LCM of $12$ and $30$ . Show that the HCF multiplied by the LCM is equal to $12 \times 30$ .
[3 marks]

C3
The LCM of $6$ and a number $n$ is $42$ . $n$ is an odd number greater than $10$ . Find $n$ .
[2 marks]
Answer
C4
Red bricks are $30$ mm tall, blue bricks are $45$ mm tall and green bricks are $50$ mm tall. Three towers are built, one from each colour, all the same height. What is the smallest height they can be? Give your answer in centimetres.
[2 marks]
Answer
C5
How many whole numbers from $1$ to $100$ are multiples of both $6$ and $8$ ?
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
This question is about the numbers $18$ and $24$ .
(a)
Find the lowest common multiple (LCM) of $18$ and $24$ .
[2 marks]
Answer
(b)
Find the highest common factor (HCF) of $18$ and $24$ .
[1 mark]
Answer
Total for D1: 3 marks

D2
Ben says, "The LCM of $10$ and $14$ is $140$ , because $10 \times 14 = 140$ ."
Ben is wrong. Explain why, and write down the correct LCM.
[2 marks]
D3
Sausages are sold in packs of $12$ . Bread rolls are sold in packs of $20$ . Tom wants to buy exactly the same number of sausages and bread rolls.
(a)
Work out the smallest number of packs of sausages and the smallest number of packs of bread rolls he can buy.
[3 marks]
Answersausages $\dots\dots$ rolls $\dots\dots$
(b)
A pack of sausages costs £2.40. A pack of bread rolls costs £1.15. Work out the cost of the packs in part (a).
[2 marks]
Answer
Total for D3: 5 marks
D4
Three warning lights flash. The first flashes every $6$ seconds, the second every $8$ seconds and the third every $10$ seconds. They all flash together at 9:00 pm.
(a)
How many seconds after 9:00 pm do they next all flash together?
[2 marks]
Answer
(b)
How many times do they all flash together in the $15$ minutes after 9:00 pm? Do not count the flash at 9:00 pm.
[2 marks]
Answer
Total for D4: 4 marks
D5
Work out $\frac{3}{4} + \frac{5}{6}$ . Give your answer as a mixed number.
[3 marks]
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Three cogs turn together in a row. Cog $A$ has $18$ teeth, cog $B$ has $30$ teeth and cog $C$ has $45$ teeth. Each cog has a dot on one tooth, and the three dots start at the top. When one tooth of a cog moves past, one tooth of each other cog moves past. How many complete turns does each cog make before all three dots are back at the top at the same time for the first time?
[4 marks]
Hint 1 · What to try
Count the teeth that move past. The dot on cog $A$ is back at the top after $18$ , $36$ , $54$ , ... teeth.
Hint 2 · The first line
You need the smallest number of teeth that is a multiple of $18$ , of $30$ and of $45$ .
Hint 3 · The full method
The LCM of $18$ , $30$ and $45$ is $90$ . So $A$ turns $90 \div 18 = 5$ times, $B$ turns $90 \div 30 = 3$ times and $C$ turns $90 \div 45 = 2$ times.
Answer $A$ $\dots$ $B$ $\dots$ $C$ $\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the lowest common multiple of two or three numbers.
I can use the LCM to add and subtract fractions.
I can use the LCM to solve problems about things that happen together again.

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